Abstract

We define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expression for the Sabban frame vectors of special pseudohyperbolic Smarandache curves. Finally, we give some examples of such curves.

1. Introduction

In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the problem of characterization of a regular curve. In the solution of the problem, the curvature functions and of a regular curve have an effective role. It is known that we can determine the shape and size of a regular curve by using its curvatures and . Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, that is, a geometry which has at least one Smarandachely denied axiom [1]. An axiom is said to be Smarandachely denied if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes.

If the position vector of a regular curve is composed by the Frenet frame vectors of another regular curve , then the curve is called a Smarandache curve [2]. Special Smarandache curves in Euclidean and Minkowski spaces are studied by some authors [3–8]. The curves lying on a pseudohyperbolic space in Minkowski 3-space are characterized in [9].

In this paper, we define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame's vectors of special pseudohyperbolical Smarandache curves. In particular, we prove that special -pseudohyperbolical Smarandache curves do not exist. Besides, we give some examples of special pseudohyperbolical Smarandache curves in Minkowski 3-space.

2. Basic Concepts

The Minkowski 3-space is the Euclidean 3-space provided with the standard flat metric given by
where is a rectangular Cartesian coordinate system of . Since is an indefinite metric, recall that a nonzero vector can have one of the three Lorentzian causal characters: it can be spacelike if , timelike if , and null (lightlike) if . In particular, the norm (length) of a vector is given by and two vectors and are said to be orthogonal if . Next, recall that an arbitrary curve in can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors are, respectively, spacelike, timelike, or null (lightlike) for every [10]. If for every , then is a regular curve in . A spacelike (timelike) regular curve is parameterized by pseudo-arclength parameter which is given by , and then the tangent vector along has unit length, that is, for all , respectively.

For any and in the space , the pseudovector product of and is defined by

Remark 1. Let , , and be vectors in . Then, (i),(ii),
(iii),
where is the pseudovector product in the space .

Lemma 2. In the Minkowski 3-space , the following properties are satisfied [10]:(i)two timelike vectors are never orthogonal;(ii)two null vectors are orthogonal if and only if they are linearly dependent;(iii)timelike vector is never orthogonal to a null vector.

Pseudohyperbolic space in the Minkowski 3-space is a quadric defined by

Let be a regular unit speed curve lying fully in in . Then its position vector is timelike vector, which implies that tangent vector is the unit spacelike vector for all . Hence we have orthonormal Sabban frame along the curve , where is the unit spacelike vector. The corresponding Frenet formulae of , according to the Sabban frame, read
where is the geodesic curvature of on in and is arc length parameter of. In particular, the following relations hold:

3. Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

In this section, we define and investigate pseudohyperbolical Smarandache curves in Minkowski 3-space according to the Sabban frame.

Let and be two regular unit speed curves lying fully in pseudohyperbolic space in and let and be the moving Sabban frames of these curves, respectively. Then we have the following definitions of pseudohyperbolical Smarandache curves.

Definition 3. Let be a regular unit speed curve lying fully in . Then -pseudohyperbolical Smarandache curve of is defined by
where and .

Definition 4. Let be a regular unit speed curve lying fully in . Then -pseudohyperbolical Smarandache curve of is defined by
where and .

Definition 5. Let be a regular unit speed curve lying fully in . Then -pseudohyperbolical Smarandache curve of is defined by
where and .

Theorem 6. Let be a regular unit speed curve lying fully in . Then -pseudohyperbolical Smarandache curve of does not exist.

Proof. Assume that there exists -pseudohyperbolical Smarandache curve of . Then it can be written as
where , which is a contradiction.

In the theorems which follow, we obtain Sabban frame and geodesic curvature of pseudohyperbolical Smarandache curve .

Theorem 7. Let be a regular unit speed curve lying fully in with the Sabban frame and the geodesic curvature . If is -pseudohyperbolical Smarandache curve of , then its frame is given by
and the corresponding geodesic curvature reads
where and .

Proof. Differentiating (6) with respect to and using (4), we obtain
where
Therefore, the unit spacelike tangent vector of the curve is given by
where if for all and if for all . Differentiating (14) with respect to , we find
and from (13) and (15) we get
On the other hand, from (6) and (14) it can be easily seen that
is a unit spacelike vector. Consequently, the geodesic curvature of the curve is given by

Theorem 8. Let be a regular unit speed curve lying fully in with the Sabban frame and the geodesic curvature . If is -pseudohyperbolical Smarandache curve of , then its frame is given by
and the corresponding geodesic curvature reads
where
and .

Proof. Differentiating (7) with respect to and using (4), we obtain
where
Therefore, the unit spacelike tangent vector of the curve is given by
Differentiating (24) with respect to , it follows that
From (4) and (23), we get
where
On the other hand, from (7) and (24), it can be easily seen that
Hence is a unit spacelike vector.Therefore, the geodesic curvature of the curve is given by

Theorem 9. Let be a regular unit speed curve lying fully in with the Sabban frame and the geodesic curvature . If is -pseudohyperbolical Smarandache curve of , then its frame is given byand the corresponding geodesic curvature reads
where
and .

Proof. Differentiating (8) with respect to and by using (4), we find
and thus
where
The geodesic curvature for all , since is a unit speed regular curve in . Therefore, the unit spacelike tangent vector of the curve is given by
Differentiating (36) with respect to and from (4) and (35), it follows that
where
On the other hand, from (8) and (36) it can be easily seen that
Hence is a unit spacelike vector.Therefore, the geodesic curvature of the curve is given by

Corollary 10. If is a geodesic curve on in Minkowski 3-space , then(1)-pseudohyperbolic Smarandache curve is also geodesic on ;(2)-pseudohyperbolic and -pseudohyperbolic Smarandache curves have constant geodesic curvatures on .

4. Examples

Example 1. Let be a unit speed curve lying in pseudohyperbolic space in the Minkowski 3-space with parameter equation (see Figure 1)
The orthonormal Sabban frame along the curve is given by
In particular, the geodesic curvature of the curve has the form

Figure 1: The curve on .

Case 1. If we take , , then from (6) the -pseudohyperbolical Smarandache curve is given by (see Figure 2)
From Theorem 7, its frame is given by
and the corresponding geodesic curvature reads

Figure 2: The -pseudohyperbolical Smarandache curve and the curve on .

Case 2. If we take , , then from (7) the -pseudohyperbolical Smarandache curve is given by (see Figure 3)
According to Theorem 8, its frame is given by
and the corresponding geodesic curvature reads

Figure 3: The -pseudohyperbolical Smarandache curve and the curve on .

Case 3. If , , and , then from (7) the -pseudohyperbolical Smarandache curve is given by (see Figure 4)
By using Theorem 9, it follows that the frame is given by (see Figure 4)
and the corresponding geodesic curvature (Figure 4) reads

Figure 4: The -pseudohyperbolical Smarandache curve and the curve on .

Also, Pseudohyperbolical Smarandache curves of and the curve on (1) with Figure 5 are shown.

Figure 5: Pseudohyperbolical Smarandache curves of and the curve on .

Acknowledgment

The authors would like to thank the referees for their helpful suggestions and comments which significantly improved the first version of the paper.